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arX
iv:0
707.
0551
v1 [
cond
-mat
.sup
r-co
n] 4
Jul
200
7
Scanning tunneling spectroscopy characterization of the
pseudogap and the x = 18 anomaly in La2−xSrxCuO4 thin films
Ofer Yuli, Itay Asulin, and Oded Millo∗
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Gad Koren
Department of Physics, Technion - Israel Institute of Technology, Haifa 32000, Israel
Abstract
Using scanning tunneling spectroscopy we examined the local density of states of thin c-axis
La2−xSrxCuO4 films, over wide doping and temperature ranges. We found that the pseudogap
exists only at doping levels lower than optimal. For x = 0.12, close to the ’anomalous’ x = 1
8doping
level, a zero bias conductance peak was the dominant spectral feature, instead of the excepted V-
shaped (c-axis tunneling) gap structure. We have established that this surprising effect cannot be
explained by tunneling into (110) facets. Possible origins for this unique behavior are discussed.
PACS numbers: 74.72.Dn, 74.50.+r, 74.78.Bz, 74.25.Jb
∗Electronic address: [email protected]
1
I. INTRODUCTION
For a certain doping range, at temperatures higher than the superconducting transition
temperature Tc, hole doped cuprate high temperature superconductors (HTSC) exhibit de-
pletion in the density of states (DOS) around the Fermi energy, EF . This ’soft gap’, known
as the pseudogap (PG) persists up to a doping dependent temperature T ∗ > Tc. The PG
phenomenon is not yet well understood and it is even still unclear whether its origin is re-
lated to superconductivity or not. Experimentally, the PG has been studied extensively for
different HTSCs, using various methods. [1] Several experiments show that the PG exists
only in the underdoped regime and that there, for each doping level, it evolves smoothly
into the superconducting (SC) gap at the corresponding Tc. This suggests that both phases
share a similar energy scale. In contrast, quite a few studies reveal deviations from the above
’universal’ behavior. Angle resolved photo emission spectroscopy (ARPES) investigations
[2, 3] imply that there may be two PGs of different origins in La2−xSrxCuO4 (LSCO), with
typical gap values of ∼ 30 and ∼ 100 meV. Scanning tunneling microscopy and spectroscopy
(STM and STS) studies of Bi-based cuprates show that the PG extends into the overdoped
regime, [4, 5] while no PG has been measured for optimally- and overdoped Y Ba2Cu3O7−δ
(YBCO). [6] These contradicting data raise the question of whether the observed results are
truly universal attributes of the PG in the HTSC cuprates.
STS enables precise measurements of the single particle excitation spectra, with high
energy and spatial resolution, making it a powerful tool for studying the energy scales
of both the SC gap and the PG. Atomically-resolved STS measurements of the DOS in
Bi2Sr2CaCu2O8−δ (BSCCO) revealed large spatial fluctuations of the quasi-particle DOS
[7, 8, 9, 10] and provided the first tunneling spectroscopy data of the PG doping and tem-
perature dependencies. [4] In fact, most of the STS investigations of the cuprates has been
performed on BSCCO due to the relative ease of achieving a clean surface by cleavage under
vacuum. In contrast, very few STM studies have been carried out on LSCO due to problems
of surface degradation. The same problem also hindered STM measurements of heavily un-
derdoped YBCO.
The relatively low Tc and ease of sample preparation over a wide doping range make
LSCO an ideal candidate for studying the PG. This is illustrated by the abundant exper-
imental data available, acquired using various techniques such as transport measurements,
2
[11] point contact spectroscopy [12], tunnel junctions, [13] infra red spectroscopy, [14] and
ARPES. [15] It is therefore unfortunate that STS measurements on LSCO are relatively
scarce, and were performed mainly on optimally doped samples. Obviously, it is very im-
portant to expand the STS measurements in LSCO also to the under- and overdoped regimes
of the phase diagram, to obtain a comprehensive picture of the doping dependence of the
DOS and elucidate the relation between the SC and PG states. This will hopefully be an
important step towards resolving the mechanism underlying high Tc superconductivity.
Another unresolved property of LSCO and related cuprates (e.g. the La-214 family) is
the x = 1
8anomaly, where a suppression of the superconducting transition temperature is
observed. This is manifested in La2−xBaxCuO4 [16] where Tc drops to nearly zero at x = 1
8,
and in LSCO, in a more subtle way, where a plateau develops in the Tc versus doping curve,
Tc(x). [17, 18] In addition, magnetization and penetration depth results have recently been
reported to behave anomalously at x = 1
8. [19] The above phenomena have been attributed
by various authors to a stripe phase known to exist in these compounds. [20] The connection
with stripes emerges from the fact that at x = 1
8the stripe spacing is commensurate with the
underlying crystal structure. We note in passing, that anomalies are expected also for other
x values at which commensurability conditions prevail, often referred to as magic numbers.
[21]
Following the above considerations, we performed a systematic STS study of LSCO over
a wide range of doping and temperature. To the best of our knowledge, this is the first
report on STS measurements of the temperature evolution of the DOS in underdoped and
overdoped LSCO, and for the x = 1
8doping level in particular. Our main findings are that
the PG in LSCO pertains only to the underdoped regime, and is absent for doping levels
higher than optimal. Additionally, for c-axis films with x = 0.12, a zero bias conductance
peak (ZBCP) was observed on large smooth c-axis areas of the sample surface, replacing
the excepted V-shaped (c-axis tunneling) gap. This is a surprising result, since the ZBCP
is known to be the dominant spectral feature on (110) surfaces due to the formation of
Andreev bound states, while in c-axis tunneling spectra no such ZBCP is expected.
3
II. EXPERIMENT
90 nm thick LSCO films were epitaxially grown on (100) SrT iO3 wafers by laser ablation
deposition, with c-axis orientation normal to the substrate. About three films for each
of the following doping levels were measured: x = 0.08, 0.10, 0.12 (underdoped); x = 0.15
(optimally doped) and x = 0.18 (overdoped). The LSCO films consisted of c-axis crystallites
having well defined facets, 50 - 100 nm long, as shown in Fig. 1(a). As will be detailed below,
most of the exposed side facets are (100) oriented, but some exposed side facets are of the
(110) surface. The samples were transferred from the growth chamber in a dry atmosphere
and introduced into our cryogenic STM after being exposed to ambient air for less than
5 min. About six tunneling spectra (dI/dV vs. V characteristics) were acquired at each
position to assure data reproducibility. We also checked the dependence of the tunneling
spectra on the voltage and current settings (i.e., the tip-sample distance, or the tunneling
resistance values, in the range of 100 MΩ to 1 GΩ) and found no effect on the measured
gap features, disproving the possibility that these gaps are even partially related to single
electron charging effects. [22] The temperature dependent resistance [R(T)] measurements
were performed using the standard 4-probe technique, where special care was taken to
stabilize the temperature before each resistance measurement and to avoid sample heating.
III. RESULTS AND DISCUSSION
A. Evolution of the gap structure with doping and temperature
For all the doping levels studied, except for x = 0.12 (see below), the tunneling spectra
measured on top of the LSCO crystallites exhibited V-shaped gap structures typical of c-
axis tunneling, further confirming the assigned crystallographic orientation found by X-ray
diffraction. A typical spectrum is presented in Fig. 1(b) along with a fit to the model for
tunneling into a d -wave superconductor. [23] In many cases, the spectra acquired on the
crystallite side facets portrayed a U-shaped gap structure [see Fig. 1(c)], signifying tunneling
into smooth (100) facets, as discussed in Ref. 24. Finally, as depicted by Fig. 1(d), along
some of the side facets, we have measured ZBCPs (usually inside a gap-like structure),
suggesting that the corresponding facet exposes the (110) surface, or at least that tunneling
4
???
nm400 nm
(a)
-20 -10 0 10 200.0
0.5
1.0
1.5
2.0
(c)
dI/
dV
(arb
. u
nit
s)
Sample bias (mV)
-30 -20 -10 0 10 20 300.5
1.0
1.5
(d)
Sample bias (mV)
dI/
dV
(arb
. u
nit
s)
(b)
Sample bias (mV)
-30 -20 -10 0 10 20 300.0
0.5
1.0
1.5
dI/
dV
(arb
. u
nit
s)
FIG. 1: (a) 400x400 nm2 topography image of x = 0.08 LSCO film. (b-d) Tunneling spectra
at 4.2 K (blue curves) taken on a single crystallite. The spectrum in (b) was measured a top
of the crystallite and shows a typical c-axis V-shaped gap structure. The spectra in (c) and (d)
were acquired on two different facets of the crystallite. The one in (c) portrays a U-shaped gap,
typical for tunneling into a smooth (100) facet, and that in (d) exhibits a ZBCP, signifying that
the tunneling took place out of the anti-nodal direction. The dotted red lines are fits to the theory
of tunneling into a d-wave SC.
takes place in the a-b (CuO2) plane in a direction different from [100]. Indeed, the dotted
red line in Fig. 1(d) was calculated using the aforementioned model [23] assuming tunneling
to the a-b plane at an angle of π
3with respect to the [100] axis. A similar correlation
between the tunneling spectra and the surface nano-morphology was observed in our previous
investigations of YBCO films. [25] The ZBCP, a unique spectral feature of d -wave SCs, [26]
will be further discussed below. Interestingly, pronounced above-gap structures appear in
the tunneling spectra that were aquired on the side facets of the c-axis crystallites [spectra
(c) and (d)]. Such features occasionally appear, in a weaker manner, also in the c-axis
(on crystallite) tunneling data. These above-gap features may be related to the phonon
structure, which may be more efficiently detected when tunneling into the a-b plane. [27]
In fig 2 we present differential conductance spectra acquired at selected temperatures
on three samples: a heavily underdoped x = 0.08 (a), an optimally doped x = 0.15 (b)
and an overdoped x = 0.18 (c), samples. The gap value, ∆, was extracted from the data
by taking half the width between the coherence peaks. At temperatures higher than 4.2 K,
5
-30 -20 -10 0 10 20 301.0
1.5
2.0
2.5
40K
35K
32K
15K
4.2K
x=0.15 (b)
Sample bias-30 -20 -10 0 10 20 30
1.0
1.5
2.0
2.5
3.0
100K80K70K
30K
15K
4.2Kx=0.08 (a)
Sample bias
0.05 0.10 0.15 0.20 0.250
10
20
30
40
8
10
12
(d)
x
∆4.
2 K
(meV
)
x=0.18 (c)
dI/d
V(a
rb.u
nits
)
Sample bias-30 -20 -10 0 10 20 30
1.0
1.5
2.0
40K
10K
20K26K31K35K
4.2K
dI/d
V(a
rb.u
nits
)
dI/d
V(a
rb.u
nits
)T
c (K
)
FIG. 2: (a-c) Tunneling dI/dV vs. V curves (shifted vertically for clarity) at various temperatures
and for three different doping levels: (a) x = 0.08 (underdoped, Tc ≈ 20 K); (b) x = 0.15 (optimally
doped, Tc ≈ 35 K); (c) x = 0.18 (overdoped, Tc ≈ 29 K). (d) The 4.2 K gap (red circles, right axis)
and Tc (black squares, left axis) as a function of doping. For comparison we plot Tc(x) measured
on LSCO single crystals from T. Matsuzaki, Phys. Chem. Sol. 62, 29 (2001).
where the coherence peaks were smeared, the maximum change of the slope was taken as the
position of the gap edge. This method, which yields some uncertainty in the determination
of the gap values, was applied before for tunneling spectra obtained on LSCO, since in many
cases the coherence peaks were undetectable. [28] Fig. 2(d) summarizes the ∆4.2K values
(red circles, right axis) and the measured Tc values (black rectangles, left axis) for all the
doping levels measured. The error bars associated with the gap data represent the range
of spatial fluctuations in the ∆4.2K values, whereas those corresponding to Tc reflect the
width of the transition. For reference we have added the Tc(x) values measured on single
crystals (asterisks and dotted line) from Ref. 17. The systematic lower Tc values measured
on our thin films, as compared to those of the single crystals, is probably due to strain
forces generated by the mismatch between the lattice constants of the SrT iO3 substrate
and the LSCO film. [18, 29] The evolution of ∆ with doping appears to follow the SC dome
structure. This appears to be a reasonable result, if the gap magnitude is related to the pair
potential of the Cooper pairs that, in turn, is associated with Tc. It should be emphasized,
however, that there are reports of STS data showing different behaviors. [4, 30] We note
in passing that our observation for the doping dependence of ∆4.2K is similar to that of
the peak in the Raman spectra associated with nodal-quasiparticles excitations, reported
6
T (K)
∆(m
eV
)
R (Ω
)
x=0.10x=0.10x=0.08x=0.08
x=0.15x=0.15 (c) x=0.18x=0.18 (d)
0 10 20 30 40 500
5
10
15
0
5
10
0 10 20 30 40 50 600
5
10
15
0
5
10
15
0 20 40 600
5
10
15
0
50
100
0 20 40 60 800
10
20
0
15
30
(b)(a)
PGPG PGPG
FIG. 3: Summary of the SC gap width evolution with temperature for underdoped samples, x =
0.08 (a) and x = 0.10 (b), optimally doped, x = 0.15 (c) and overdoped, x = 0.18 (d) samples.
The gaps (black circles, left axes) are plotted along with the R(T) data (blue squares and lines,
right axes) for each doping level. The double sided arrow in (a) and (b) marks the temperature
range where the PG was clearly detectable as a depression in the tunneling spectra. At the right
hand side of the double sided arrow (T ∗) the zero-bias conductance increases toward the normal
conductance and consequently the gap structure vanishes. These data show that the PG exists
only for underdoped samples.
in Ref 31. The origin of this correspondence is not yet clear to us. Moreover, we do not
find evidence for the two energy scales discussed in the latter paper, maybe because c-axis
tunneling spectroscopy probes simultaneously the anti-nodal and nodal excitations.
Fig 3 summarizes the evolution with temperature of the SC gap width (below Tc)
for different doping levels. The gap values (black circles, left axes) were extracted from
the differential conductance curves, as explained above. The error bars represent standard
deviations of the spatial distribution of the gap magnitude and the error in determining the
gap value when the coherence peaks are absent, as discussed above. For each doping level we
present the corresponding R(T) curve that was measured right after the STM measurements.
As seen in Fig. 2(a), the SC gap of the underdoped x = 0.08 films does not close up at
Tc, as a BCS gap would. Instead, the gap evolved smoothly into a PG, manifested by a
depletion in the density of states around EF . The PG that was noticeable up to T ∗ ∼ 70
K, well above the onset temperature of superconductivity, T onset
c∼ 30 K. The smearing of
the gap edges at T > Tc created large errors in the determination of the (pseudo)gap width,
7
therefore we do not plot the gap values. Nevertheless, the temperature range where the PG
was clearly detectable is presented in Fig. 3(a) as a double sided arrow, starting at Tc and
ending at the temperature, T ∗, where the spectra turned gapless (or Ohmic). Qualitatively,
a similar behavior was observed for the (still underdoped) x = 0.10 samples, with T ∗ ∼ 50
K [Fig. 3(b)].
The results of the optimally doped and overdoped films are shown in Figs. 3(c) and 3(d).
As in the case of the underdoped samples, the SC-gap width remains almost constant nearly
up to Tc. However, in contrast with the underdoped films, the gap in the DOS closes up
and vanishes near Tc, while no sign of the PG is observed. The behavior of the overdoped
(x = 0.18) differs even more profoundly from that of the underdoped samples. As can be seen
in Figs. 2(c) and 3(d), the width of the SC gap (as well as its depth) reduced significantly
already at ∼ 2
3T onsetc , and completely disappears at T onset
c ∼ 31 K. This behavior resembles
the temperature evolution of a BCS-like gap. [32] Our data may thus portray a crossover
from a non-BCS to a BCS-like SC gap behavior as the doping increases from below to above
optimal doping, as suggested previously. [33, 34]
We conclude that our STS results corroborate earlier findings on LSCO (using other
experimental methods) as well as on other cuprates, showing that the PG is an intrinsic
property of the underdoped regime. However, both the doping dependence of the gap
magnitude and the restriction of the PG to the underdoped regime exclusively, are in contrast
to findings by Renner et al. for BSCCO. [4] These authors found clear evidence for a PG in
overdoped BSCCO samples (while we do not) and reported that ∆ monotonically decreased
as the doping level was increased (whereas we found that ∆ follows the SC dome, see Fig. 2).
These discrepancies lead one to question the universality of the SC gap and PG attributes
among the family of the HTSC cuprates. We note that the cuprates differ from one another
in various ways, e.g., in the number of CuO2 planes within the unit cell, the Tc values and
the doping methods, thus universality should not necessarily prevail.
B. Anomalous behavior at x = 0.12
When examining c-axis LSCO films with x = 0.12 (≈ 1
8) we encountered a surprising
effect. The local tunneling spectra over large areas of the sample, including those measured
on top of c-axis crystallites, exhibited ZBCPs instead of the expected c-axis V-shaped
8
Bias (mV)
100 nm
dI/
dV
FIG. 4: Tunneling spectra measured at 4.2 K on a c-axis x = 0.12 LSCO film. The spectra were
taken sequentially at equidistant steps (∼ 4 nm) along the 100 nm long line running on top of a
c-axis crystallite, as shown in the inset. Surprisingly, the spectra exhibit a ZBCP instead of a pure
V-shaped gap structure.
tunneling gap. In Fig. 4 we present a typical line scan taken on top of a c-axis crystallite
(shown in the inset) in one of our x = 0.12 films. All the spectra indeed exhibit clear
ZBCP structures. This behavior was observed for all three x = 0.12 samples we measured,
on different locations on each film separated by a few millimeters from one another and
for various directions of the line-scan. Areas showing the expected (ZBCP-free) gap
structure were also found, but they comprised less than 15% of the scanned sample area
[the corresponding data were used in the analysis shown in Fig. 2(d)]. We would like
to emphasize that the unexpected appearance of the ZBCP was exclusively found in the
x = 0.12 doping level, whereas for other doping levels only V-shaped gaps were observed on
top of the crystallites (see Fig. 1). In the latter (x 6= 0.12) films, the ZBCPs were detected
only along some of the crystallite facets, as discussed above.
The ZBCP spectral feature is one of the hallmarks of d -wave superconductivity.
[23, 26] It originates from the formation of zero-energy (at EF ) surface bound states,
known as Andreev bound states (ABS). The ABS appear on the nodal surfaces of d -wave
superconductors [(110) in LSCO and YBCO] due to constructive interference of multiple
Andreev reflected quasiparticles that experience a sign change of the order parameter in
consecutive reflections. [23, 26] The ZBCP is not expected to be observed in spatially
9
-5 0 5
0.0
0.5
20
29 K
10
4.2 K
-20 -10 0 10 20
1.0
1.5
2.0
dI/
dV
(ar
b.
un
its)
Sample bias (mV)
10
29
4.2 K
32 K
20
15
FIG. 5: Tunneling spectra curves at various temperatures measured on an x = 0.12 LSCO film.
Inset: The ZBCPs after background subtraction.
resolved STS data acquired on top of c-axis crystallites, but can be detected on side facets
of these crystallites, as demonstrated above and even more clearly on pure (110) facets. [25]
We believe that the anomalous abundance of ZBCPs reported here for the x = 0.12 films
cannot be associated with faceting and local tunneling along the nodal [110] direction (in
spite of the fact that they look very similar to the nodal, ABS-related, ZBCPs). As noted
above, line scans continuously showing ZBCPs were measured along different directions
(that did not necessarily run along any major crystallographic direction) on a single
crystallite. In addition, no ZBCP were found on-top of crystallites in the films of other
(x 6= 0.12) doping levels, indicating that the presence of (110) nano-facets is negligible in
our films. These findings practically rule out the possibility that the ZBCPs are associated
with (110) facets. Moreover, X-ray diffraction measurements on our samples revealed highly
oriented c-axis films with sharp (00n) peaks and no other orientations.
Next we address the temperature evolution of the ZBCP in the x = 0.12 films. As
demonstrated in Fig. 5, the magnitude of the ZBCP gradually decreases as the temperature
is raised and vanishes at about 30 K, very close to (and slightly above) the SC transition
temperature. In order to compare the ’anomalous’ ZPCPs with the ’conventional’ ones
associated with the nodal ABS, we have performed temperature dependent STS measure-
ments on x = 0.12 LSCO films grown intentionally on (110) STO wafers. X-ray diffraction
measurements showed that in addition to the (110) LSCO orientation, a large amount of
(103) orientation is also present in these films. This however posed no problem since, as
10
previously shown, ZBCPs due to ABS exist also for this orientation. [35, 36] Fig. 6 displays
a comparison between the temperature evolution of the ZBCP magnitude measured on
(001) and nominal (110) films. Each data point in this figure represents the integrated area
under the corresponding ZBCP after subtraction of the background conductance as shown
in the inset of Fig. 5. Assuming that the ZBCP observed on the c-axis film is also due
to some type of zero-energy surface bound states, the data in Fig. 6 reflect the number of
bound states at each temperature. [37] Both sets of data show that the ZBCP magnitude
decreases monotonically as the temperature is increased. There is, however, a marked
difference between the two samples, namely, the two types of ZBCPs. While the ZBCPs
on the (001) surface were still detectable at around Tc ≈ 27K, the ’conventional’ ZBCPs
vanished already at a much lower temperature, ∼ 18 K, well below Tc.
Previous tunneling spectroscopy investigations of the nodal surfaces of LSCO and
YBCO showed that the ABS usually lose all spectral weight at about 0.5 − 0.6Tc. [13, 38]
In particular for the (110) surface of an x = 0.12 LSCO single crystal, Dagan et al. have
shown, using point contact spectroscopy, that the ZBCP disappeared at ∼ 15 K, while Tc
was ∼ 31 K. The qualitative agreement of this finding with our results for the nominal (110)
surface [Fig. 6(b)] on one hand, and the different behavior we found for the ’anomalous’
(001)-surface ZBCPs [Fig. 6(a)] on the other hand, further support our claim that the
x = 0.12 anomaly is not a faceting effect. Nevertheless, the fact that the ’anomalous’
ZBCPs have a similar shape to that of the nodal ones and that they also quench with
increasing temperature suggests that they also manifest surface bound states resulting
from multiple Andreev reflections. The exact mechanism (or multiple Andreev reflection
process) that underlines the formation of these bound states is probably different from
that associated with the nodal ABS [26] and has not been predicted or discussed in the
literature as of yet.
Bobkova et al. predicted theoretically that zero-energy bound states should exist at
the interface between the a-b plane of a d -wave SC and a charge density wave (CDW)
material, provided this junction has high enough transparency. [39] These bound states
represent a combined effect of Andreev reflections from the superconducting side, un-
conventional Q-reflections from the CDW solid, and standard specular reflections. In a
Q-reflection from the interface with the CDW solid, a sub CDW-gap quasiparticle changes
its momentum by the wavevector Q of the CDW pattern. While this model is not directly
11
0 10 20 30 40 50
0.0
0.5
1.0
1.5
2.0
Tc
(110)(110)
(b)
(001)(001)
(a)
Tc
Inte
gra
ted
are
a o
f Z
BC
P (
arb
. u
nit
s)
T (K)
0 10 20 30 40 500.0
0.5
1.0
1.5
2.0
2.5
FIG. 6: The temperature dependence of the spectral weight the ZBCPs measured on the (001)
(a) and (110) (b) films. The (001) ZBCP vanished at around Tc, whereas that measured on the
(110) film vanishes at a lower temperature, as commonly found for ZBCPs associated with the
’conventional’ nodal ABS.
related to our experimental configuration, it may indicate a possible direction at which a
relevant model could be established, namely, the involvement of static stripes. Neutron
scattering measurements have shown the existence of static stripes commensurate with
the underlying crystal structure in LSCO for x = 1
8. [20] These stripes exhibit charge
modulation that can be related to a CDW order parameter, which coexists with a d -wave
order parameter. A serious theoretical investigation is thus needed on the possible role
of stripes in the appearance of the ZBCP anomaly that we observed in the x = 1
8LSCO films.
12
IV. SUMMARY
In conclusion, we have shown that the PG in epitaxial thin films of LSCO exists only
in the underdoped regime where above Tc, the gap became more and more shallow and
eventually vanished at a doping dependent temperature T ∗ > Tc . At 4.2 K the SC gap
magnitude was found to qualitatively follow the SC dome structure versus doping, as
expected for a gap magnitude proportional to the pairing potential. The disagreement of
our results with those obtained on BSCCO suggests that the SC gap and PG properties
may not be universal among the HTSC cuprates. We have also revealed a new anomaly
associated with the special x = 1
8doping level of LSCO. The c-axis tunneling spectra
exhibited a predominant ZBCP instead of the expected V-shaped gap, manifesting the
creation of a new type of zero energy bound states. Ruling out the possibility that this is
a facet effect, we conjecture that this anomalous spectral feature is associated with static
stripes. Further theoretical modeling and experimental data using different methods are
needed in order to resolve the puzzle associated with this intriguing effect.
V. ACKNOWLEDGMENTS
The authors are grateful to G. Deutscher, D. Orgad, and S. Baruch, for helpful and
stimulating discussions. This research was supported by the Israel Science Foundation,
Center of Excellence program (grant # 1565/04).
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